Showing papers on "Prim's algorithm published in 2001"

Minimum cost spanning tree games and population monotonic allocation schemes

Abstract: In this paper we present the Subtraction Algorithm that computes for every classical minimum cost spanning tree game a population monotonic allocation schemeAs a basis for this algorithm serves a decomposition theorem that shows that every minimum cost spanning tree game can be written as nonnegative combination of minimum cost spanning tree games corresponding to 0-1 cost functionsIt turns out that the Subtraction Algorithm is closely related to the famous algorithm of Kruskal for the determination of minimum cost spanning treesFor variants of the classical minimum cost spanning tree games we show that population monotonic allocation schemes do not necessarily exist

Approximating the Minimum Spanning Tree Weight in Sublinear Time

Abstract: We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of maximum degree d, with edge weights in the set {1; ... ; ω}, and given a parameter 0 < Ɛ < 1/2, estimates in time O(dωƐċ2 log ω/Ɛ) the weight of the minimum spanning tree of G with a relative error of at most Ɛ. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dωƐċ2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dƐċ2 logƐċ1) the number of connected components of an unweighted graph to within an additive error of ∈n. The time bound is shown to be tight up to within the log Ɛċ1 factor. Our connected-components algorithm picks O(1/∈2) vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST.

Self-Stabilizing Minimum Spanning Tree Construction on Message-Passing Networks

Abstract: Self-stabilizing algorithms for constructing a spanning tree of an arbitrary network have been studied for many models of distributed networks including those that communicate via registers (either composite or read/write atomic) and those that employ message-passing. In contrast, much less has been done for the corresponding minimum spanning tree problem. The one published self-stabilizing distributed algorithm for the minimum spanning problem that we are aware of [3] assumes a composite atomicity model. This paper presents two minimum spanning tree algorithms designed directly for deterministic, message-passing networks. The first converts an arbitrary spanning tree to a minimum one; the second is a fully self-stabilizing construction. The algorithms assume distinct identifiers and reliable fifo message passing, but do not rely on a root or synchrony. Also, processors have a safe time-out mechanism (the minimum assumption necessary for a solution to exist.) Both algorithms apply to networks that can change dynamically.

The Directed Minimum-Degree Spanning Tree Problem

Abstract: Consider a directed graph G = (V, E) with n vertices and a root vertex r ∈ V. The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasi-polynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is at most O(Δ* + log n) where, Δ* is the degree of some optimal tree for the problem. The running time of the algorithm is shown to be O(nO(log n)). Experimental results are presented showing that the actual running time of the algorithm is much smaller in practice.

Extending greedy multicast routing to delay sensitive applications

Abstract: Given a weighted undirected graph G(V,E) and a subset R of V, a Steiner tree is a subtree of G that contains each vertex in R We present an online algorithm for finding a Steiner tree that simultaneously approximates the shortest path tree and the minimum weight Steiner tree, when the vertices in the set R are revealed in an online fashion This problem arises naturally while trying to construct source-based multicast trees of low cost and good delay The cost of the tree we construct is within an O(log |R|) factor of the optimal cost, and the path length from the root to any terminal is at most O(1) times the shortest path length The algorithm needs to perform at most one reroute for each node in the tree Our algorithm extends the results of Khuller etal and Awerbuch etal, who looked at the offline problem We conduct extensive simulations to compare the performance of our algorithm (in terms of cost and delay) with that of two popular multicast routing strategies: shortest path trees and the online greedy Steiner tree algorithm

A Very Fast (Linear Time) Distributed Algorithm, on General Graphs, for the Minimum-Weight Spanning Tree.

Abstract: This paper develops linear time distributed algorithm, on general graphs, for the minimum spanning tree, in asynchronous communication network. We concentrated our efforts on the improvement of the execution time this is why our algorithm is faster than all previous linear algorithms. Our algorithm propose a solution for computing the MST in time $n/2$ with $O(n^2)$ messages (where $n=|V|$ for a graph $G=(V,E)$). The total number of messages in the worst case is slightly higher than the others algorithms, but in practice is often better.

In order to form a more perfect union [minimum spanning tree algorithm]

Abstract: The authors present an algorithm for finding a minimum spanning tree. They show a typical application of this MST algorithm. Suppose we have 11 data points in a plane and would like to find some structure among them. Suppose we draw dotted lines between each pair of points. If we consider the points vertices and the dotted lines edges, we have a complete graph on 11 points. If we then choose just enough of the edges to keep the graph connected, this is a spanning tree of the graph. A graph can have many spanning trees. The one presented is an MST, that is, a spanning tree that minimizes the total length of edges. As is explained, the most difficult part of this algorithm is implementing set UNIONs.